There are 4 routes between Delhi and Patna. In how many different ways can a man go from Delhi to Patna and return, if for returning
(i) any of the routes is taken,
(ii) the same route is taken,
(iii) the same route is not taken?

Explanation

We have the following three cases.
Case(i) When any of the routes is taken for returning:
The man may take may take any route for going from Delhi to Patna.
So, there are 4 ways of going from Delhi to Patna.
When done so, he may return by any of the 4 routes.
So, there are 4 ways of returning from Patna to Delhi.
Hence, by the foundamental principle of multiplication, the total number of way going to Patna and returning back to Delhi `= (4xx4)=16.`
Case (ii) When the same route is taken for returning:
In this case, there are 4 ways of going to Patna and only 1 way of returning, namely by the same route.
Hence, the required number of ways `= (4xx1) =4. `
Case (iii) When the same route is not taken for returning:
In this case, there are 4 ways of going to Patna.
But, the man does not return by the same route.
So, there are 3 ways of returning back to Delhi.
Hence, the required number of ways `= (4xx3) = 12.`